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Theorem cofunexg 5680
Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cofunexg ((Fun A B 𝐶) → (AB) V)

Proof of Theorem cofunexg
StepHypRef Expression
1 relco 4762 . . 3 Rel (AB)
2 relssdmrn 4784 . . 3 (Rel (AB) → (AB) ⊆ (dom (AB) × ran (AB)))
31, 2ax-mp 7 . 2 (AB) ⊆ (dom (AB) × ran (AB))
4 dmcoss 4544 . . . . 5 dom (AB) ⊆ dom B
5 dmexg 4539 . . . . 5 (B 𝐶 → dom B V)
6 ssexg 3887 . . . . 5 ((dom (AB) ⊆ dom B dom B V) → dom (AB) V)
74, 5, 6sylancr 393 . . . 4 (B 𝐶 → dom (AB) V)
87adantl 262 . . 3 ((Fun A B 𝐶) → dom (AB) V)
9 rnco 4770 . . . 4 ran (AB) = ran (A ↾ ran B)
10 rnexg 4540 . . . . . 6 (B 𝐶 → ran B V)
11 resfunexg 5325 . . . . . 6 ((Fun A ran B V) → (A ↾ ran B) V)
1210, 11sylan2 270 . . . . 5 ((Fun A B 𝐶) → (A ↾ ran B) V)
13 rnexg 4540 . . . . 5 ((A ↾ ran B) V → ran (A ↾ ran B) V)
1412, 13syl 14 . . . 4 ((Fun A B 𝐶) → ran (A ↾ ran B) V)
159, 14syl5eqel 2121 . . 3 ((Fun A B 𝐶) → ran (AB) V)
16 xpexg 4395 . . 3 ((dom (AB) V ran (AB) V) → (dom (AB) × ran (AB)) V)
178, 15, 16syl2anc 391 . 2 ((Fun A B 𝐶) → (dom (AB) × ran (AB)) V)
18 ssexg 3887 . 2 (((AB) ⊆ (dom (AB) × ran (AB)) (dom (AB) × ran (AB)) V) → (AB) V)
193, 17, 18sylancr 393 1 ((Fun A B 𝐶) → (AB) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  wss 2911   × cxp 4286  dom cdm 4288  ran crn 4289  cres 4290  ccom 4292  Rel wrel 4293  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by:  cofunex2g  5681
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